JOURNAL OF ALGEBRA 89, 148-177 (1984) A Characterization of Finite Auslander-Reiten Quivers KIYOSHI IGUSA * Brandeis University, Waltham, Massachusetts 02154 AND GORDANA TODOROV Northeastern University, Boston, Massachusetts 02115 Communicated by Walter Feit Received February 1, 1983 Let II be an indecomposableartin algebra of finite representation type, R the center of A, k = R/rad R. Let r,, be the associated Auslander-Reiten quiver. Then r,, is finite k-modulated translation quiver and it is unique up to isomorphism of modulated translation quivers (Section 2). The main results of this paper are: (a) We give necessary and sufficient conditions for a finite k-modulated translation quiver to be an Auslander-Reiten quiver in terms of certain homology groups associatedto the quiver (Section 3). (b) We show that whether a finite k-modulated translation quiver is an Auslander-Reiten quiver dependsonly on the underlying valued translation quiver and the characteristic of k and give necessary and sufficient conditions only in terms of the valued quiver and char k. We also show that any valued quiver with valuations (1, n) or (n, l), n Q 3, admits a modulation over any prime field (Sections 5,6, 7). (c) To a valued translation quiver r we associate an unvalued tran- slation quiver r (i.e., all valuations are 1) and show that a finite k-modulated translation quiver is an Auslander-Reiten quiver if and only if the associated unvalued translation quiver with the trivial K-modulation (K = prime field of k) is an Auslander-Reiten quiver (Section 8). * Research supported by NSF Grant MCS-82-02246. This author is a Sloan Fellow. 148 0021-8693184 $3.00 Copyright 0 1984 by Academic Press, Inc. All rights of reproduction in any form reserved.